#508559

order	512
nilpotency class	4
isoclinism	isoclinic class of #508494
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

417 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 417 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,3)(6,7)(9,10)(11,15)(12,23)(13,18)(14,16)(17,27)(19,35)(20,31)(21,40)(22,33)(24,30)(25,37)(26,36)(28,38)(29,34)(32,39);(1,2,5,3)(4,6,8,7)(9,11,13,28,14,20,29,22)(10,15,18,38,16,31,34,33)(12,24,26,21,27,19,25,39)(17,35,37,32,23,30,36,40);(1,3,5,2)(4,6,8,7)(11,19)(12,25)(13,29)(15,30)(17,36)(18,34)(20,24)(21,28)(22,39)(23,37)(26,27)(31,35)(32,38)(33,40);(1,4,5,8)(2,6,3,7)(10,16)(15,31)(17,23)(18,34)(30,35)(32,40)(33,38)(36,37)

Polycyclic presentation:

2^9|b^2=fh,c^2=e,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=de,g^c=ghi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415755);